Note on Locally Conformal Kähler Surfaces

The purpose of this note is to show that the complex two-dimensional locally conformal Kähler solvmanifold obtained by L. de Andres, Fernandez, Mencia and Cordero is holomorphically homothetic to the Inoue surface equipped with the locally conformal Kähler structure constructed by Tricerri. In order to prove it, we collect several facts related to the existence of locally conformal Kähler structure on compact complex surfaces.